Question: $\dfrac{ 4d + 7e }{ -7 } = \dfrac{ 2d + 2f }{ -9 }$ Solve for $d$.
Answer: Multiply both sides by the left denominator. $\dfrac{ 4d + 7e }{ -{7} } = \dfrac{ 2d + 2f }{ -9 }$ $-{7} \cdot \dfrac{ 4d + 7e }{ -{7} } = -{7} \cdot \dfrac{ 2d + 2f }{ -9 }$ $4d + 7e = -{7} \cdot \dfrac { 2d + 2f }{ -9 }$ Multiply both sides by the right denominator. $4d + 7e = -7 \cdot \dfrac{ 2d + 2f }{ -{9} }$ $-{9} \cdot \left( 4d + 7e \right) = -{9} \cdot -7 \cdot \dfrac{ 2d + 2f }{ -{9} }$ $-{9} \cdot \left( 4d + 7e \right) = -7 \cdot \left( 2d + 2f \right)$ Distribute both sides $-{9} \cdot \left( 4d + 7e \right) = -{7} \cdot \left( 2d + 2f \right)$ $-{36}d - {63}e = -{14}d - {14}f$ Combine $d$ terms on the left. $-{36d} - 63e = -{14d} - 14f$ $-{22d} - 63e = -14f$ Move the $e$ term to the right. $-22d - {63e} = -14f$ $-22d = -14f + {63e}$ Isolate $d$ by dividing both sides by its coefficient. $-{22}d = -14f + 63e$ $d = \dfrac{ -14f + 63e }{ -{22} }$ Swap signs so the denominator isn't negative. $d = \dfrac{ {14}f - {63}e }{ {22} }$